ELSEVIER Physica B 206 & 207 (1995) 154-156

Dynamical properties of the Kondo lattice

Th. Pruschke*, B. Steininger, J. Keller

lnstitut fiir FestkOrperphysik, Universitdt Regensburg, 93040, Regensburg, Germany

Abstract

The periodic version of Kondo's s-d exchange model is studied within the local approximation with NCA techniques. We present results for the local T-matrix and the local conduction-band density of states (DOS) in the different physical regimes of the model. We find an insulating gap for antiferromagnetic exchange in the particle-hole symmetric model, while off particle-hole symmetry the system appears to be metallic. For ferromagnetic interaction no apparent changes compared with the non- interacting case are found. For antiferromagnetic exchange coupling we compare our results with the behaviour typically found in heavy-fermion systems.

The theoretical description of correlated electron systems has been a challenge for almost 30 years. While a reliable approximate and partially even exact calculation of thermodynamic and dynamic quantities for impurities has been achieved during the past decade, a similar progress for concentrated systems is still not possible in general. Recently, however, a novel approach to concentrated, highly correlated materials has been proposed that is based on the basically local electron-electron interaction in models used to describe these systems [1]. The one-particle self-energy due to the local interaction is approximate- ly replaced by a purely local, i.e. k-independent one. In this local approximation one can again map the whole problem onto an impurity system where the presence of the lattice comes in through an effective, self-consistently determined bath that couples to the impurity [2].

This method has been successfully applied to the one-band Hubbard model [3], which seems appro- priate for transition-metal or even for some light- actinide compounds, where the strongly interacting 3d- or 5f-electrons may form a tight-binding band. The situation is definitely different for heavy fermion materials. Here a localized orbital (usually of f-charac-

* Corresponding author.

ter) couples through a weak hybridization to a band of quasi-particles. These systems are described by either the periodic Anderson model (PAM) or the periodic Kondo model (PKM). The latter lacks magnetic correlations between different sites and will thus be suitable only when no magnetic order is present. The localized orbitals are modeled by spins that scatter the band electrons through a local exchange interaction. Using standard notation the Hamiltonian reads:

HKondo = Z "kC+k~Ck~ - J Z Z (S, - q(~)c:.c,~ . (1) k~ i a,O

In the local approximation the self-energy of the band states is assumed to be local. The solution of the model (1) is then obtained by solving the corre- sponding impurity Kondo model with an effective conduction band described by a Green's function

~3(z) = [G,o'c(z) + X(z)l ' , (2)

where X(z) is the one-particle self-energy and G~oc(Z ) the true local band-electron Green's function. The self-consistency cycle is closed by noting that

Gloc(Z ) = ~ ( z ) -~- (4~(z)T(z)~(z), ( 3 )

with the local T-matrix T(z). The missing link is a prescription to obtain the local

0921-4526/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0921-4526(94)00392-0

Th. Pruschke et al. / Physica B 206 & 207 (1995) 154-156 155

T-matrix given an arbitrary conduction-band DOS, ~/(to) = ~ : m ~ ( ~ o - i0 ÷). Currently, we use an exten- sion of a perturbational scheme proposed by Maekawa [4]. In this earlier work the authors simply replaced the local exchange term according to

J Z -J E ctl3 fro"

with the constraint E~XI,, :~ = 1. Due to the prop- erties of the Hubbard operators this replacement violates the part ic le-hole symmetry present in the model (1) for a part icle-hole symmetric conduction band. To restore this symmetry one must use the mapping

J Z (s. a/3

- J EfXl ,oCo:*o.Xo, , + ~ t r '

instead, where J = J/4. This interaction term can be treated with the resolvent method and an extension of the non-crossing approximation (NCA) [5]. However, from the structure of the T-matrix one is forced to take more complicated diagrams into account. The resulting equations are then equivalent to those for the NCA for the single impurity Anderson model with finite Coulomb parameter [6].

The results for the local T-matrix and the local band DOS are shown in Fig. 1. The calculations were done for an unperturbed DOS A0(to ) = N e O ( D - Ito-/~1). J was chosen as IJI = 1.5D, which corresponds to a T K = 0.005D for antiferromagnetic coupling, J < 0. Fig. la collects the results for ferromagnetic coupling, J > 0. The T-matrix shows neither for the impurity (upper left part) nor for the lattice (upper right part) any pronounced structure and temperature variation, as expected. The same holds for the local band DOS (lower part) , which is nearly unaltered from its un- perturbed form. For most physical properties this means that one will find results similar to the non- interacting case. The situation is very different for antiferromagnetic coupling, J < 0 (Fig. lb) . Here the T-matrix exhibits a pronounced resonance (Ab- r ikosov-Suhl resonance) for low temperatures. At the same time a gap appears in the band DOS at the position of the resonance in the T-matrix. The depen- dence of the position of this gap on the chemical potential /x is shown in Fig. 2: for part icle-hole symmetry it lies exactly a t /x , while it is shifted away from /z for a band filling n ~ 1. In addition, the resonance and gap build up faster in the lattice as compared with the impurity. This becomes evident

- 0 . 8

0 . 5

,•/0.25 Z

0.5

¢0/D

-0.4 0 0.4 0.8 -0.8

J>0, impurity,

c0/D

-0,4 0 0.4 0.8

J~O, lattice

0

J>O. Iltttic¢ 1

0 ' ' i , , . . . . . i , i , i , J 0 -0.8 -o.4 0 0.4 0.8 -0.8 -0.4 0 0.4 0.8

,,to (a) ~o/0

0.5

0.25

0.5

0.5

~ 0.25

tD/D tolD

-0.8 -0.4 0 0.4 0.8 -0.8 -0.4 0.4 0.8 0.5

J<0. impurity [ - - T - -

, i i i i

-0.8 -0.4 0 0'.4 0.8 -0.8 a~D (b)

J<0, lattice i t

1 0.25

0

1<0, lattice 1

J 1

0.5

i i 0 -0.4 014 0.8

t~/D

0.5

Fig. 1. (a) Local T-matrix and conduction-band DOS for the Kondo model with ferromagnetic coupling (J > 0) for two different temperatures T= D/50 (full lines) and T= D/700 (broken lines). (b) The same quantities for antiferromagnetic coupling (J < 0) at the same temperatures.

from Fig. 3, where we show an enlarged view of the T-matrix near the Fermi level for a temperature T = D/700: in the lattice the ASR is higher and has a slightly larger width, This may be interpreted as a larger low-energy scale for the lattice. However, from the figure it is also clear that it is only a moderate enhancement compared with the impurity scale, defi- nitely not an order of magnitude.

In conclusion we reported the first calculations of dynamic quantities for the Kondo lattice using the local approximation for the band-electron self-energy. The results show the expected physical behaviour: no

156 Th. Pruschke et al. / Physica B 206 & 207 (1995) 154-156

1.25

If 0.75

0.5

0.25

- - n ~ l

. . . . . n = l

0 -0.01 -0.005 0.005

(o-~t)/D 0.01

Fig. 2. Conduction-band DOS for the lattice Kondo model (J <0) for a band filling n c = 1 (broken line) and a filling n c = 0.8 (full line) at a temperature T = D/700. The gap in the DOS is shifted away from the chemical potential when n c ~ l .

0.5

4 0.25

Z

- - lattice

. . . . . impurity

-0.05 -0.025 0 0.025 0.05

0IdD

Fig. 3. Comparison of the ASR for the impurity case (broken line) and the lattice model (full line) at a tempera- ture T = D/700. The resonance inthe lattice is clearly higher and broader as compared with the impurity, meaning an enhanced low-energy scale for the lattice.

qualitative changes in the band D O S for ferromagnet ic coupling (J > 0) and a rather structureless and tem- perature independent local T-matrix. There also ap- pear to be no visible changes when going from the impurity to the lattice. For ant i ferromagnet ic ex- change (J < 0) we find a strongly tempera ture depen- dent resonance in the local T-matrix at or close to the chemical potential . For a par t ic le -hole symmetric conduction band this leads to a gap at /z, i.e. the system will be an insulator for low temperatures . Away from par t ic le -hole symmetry resonance and gap are shifted away f rom/x , the system becomes metallic. Compared with the impurity we also find a modera te enhancement of the low-temperature energy scale.

References

[1] W. Metzner and D Vollhardt, Phys. Rev. Lett. 62 (1989) 324; E. Miiller-Hartmann, Z. Phys. B 74 (1989) 507.

[2] V. Jani~ and D. Vollhardt, Int. J. Mod. Phys. B 6 (1992) 731; M. Jarrell, Phys. Rev. Lett. 69 (1992) 168; A. Georges and G. Kottiar, Phys. Rev. B 45 (1992) 6479.

[3] M. Jarrell and Th. Pruschke, Z. Phys. B 90 (1993) 187. [4] S. Maekawa, S. Tajahashi, S. Kashiba and M. Tachiki, J.

Phys. Soc. Japan 54 (1985) 1955; R. Freytag and J. Keller, Z. Phys. B 85 (1991) 86; F.B. Anders, Q. Qin and N. Grewe, J. Phys.: Condens. Matter 4 (1992) 7229.

[5] H. Keiter and G. Morandi, Phys. Rep. 109 (1984) 227; N.E. Bickers, D.L. Cox and J.W. Wilkins, Phys. Rev. B 36 (1987) 2036.

[6] Th. Pruschke and N. Grewe, Z. Phys. B 74 (1989) 439.